1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

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Convex bodies : the Brunn-Minkowski theory

Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations, but their value is rather limited in genuinely non-elliptic problems. In this paper we shall therefore discuss some more general classes of operators which are adapted to such applications. For these operators we shall develop a calculus which is almost as smooth as that of pseudo-differential operators. It also seems that one gains some more insight into the theory of pseudo-differential operators by considering them from the point of view of the wider classes of operators to be discussed here so we shall take the opportunity to include a short exposition.

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Fourier Integral Operators. I

The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect9 functional provided due account is taken of its gauge symmetry. This enables topological conclusions to be drawn about the critical sets and leads eventually to information about the moduli space of algebraic bundles over the Riemann surface. This in turn depends on the interplay between the holomorphic and unitary structures, which is analysed in detail.

The Yang-Mills equations over Riemann surfaces

This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

I. Hilbert Schemes and Chow Varieties.- II. Curves on Varieties.- III. The Cone Theorem and Minimal Models.- IV. Rationally Connected Varieties.- V. Fano Varieties.- VI. Appendix.- References.

Rational Curves on Algebraic Varieties

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971."

Groupes et algèbres de Lie

Éléments de mathématique

Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce livre est le cinquieme du traite; il est consacre aux bases de l'analyse fonctionnelle. Il contient en particulier le theoreme de Hahn-Banach et le theoreme de Banach-Steinhaus. Il comprend les chapitres: -1. Espaces vectoriels topologiques sur un corps value; -2. Ensembles convexes et espaces localement convexes; -3. Espaces d'applications lineaires continues; -4. La dualite dans les espaces vectoriels topologiques; -5. Espaces hilbertiens (theorie elementaire). Il contient egalement des notes historiques. Ce volume a ete publie en 1981.

Nicolas Bourbaki

Papers

Groupes et algèbres de Lie

Éléments de mathématique

Espaces vectoriels topologiques

Lie Groups and Lie Algebras : Chapters 1-3

Eléments de mathématique : groupes et algèbres de Lie